Block Boundary Value Methods For Several Classes Of Nonlinear Functional–differential Algebraic Equations

Functional differential algebraic equations are a class of complicated equations coupled with functional differential equations and algebraic equations,which are also called as functional differential and functional equations.In the field of automatic control,this type of equations is also named as nonlinear hybrid systems with delay.In particular,neutral differential equations are covered by this type of equations.In addition,this class of equations has extremely wide applications in physics?analog chemistry?power and circuit analysis?multibody dynamics?biomedicine?automatic control?materials science?finance and other scientific fields.Compared with differential algebraic equations without delay,functional differential algebraic equations can more accurately characterize development law of objective things in nature.Generally speaking,it is challenging to derive exact solutions of this class of equations.Hence,in order to keep high approximation to the corresponding exact solutions,we have to turn to developing some highly efficient numerical algorithm to derive the numerical solution.Up to now,there are only few results about the numerical solution of functional differential algebraic equations with constant delay,including linear multistep method?one-leg method?Runge–Kutta method?general linear multistep method.However,linear multistep method has high accuracy but lack of excellent stability,and Runge–Kutta method share both high accuracy and good stability but its computational cost is expensive in solving implementation.In fact,in the numerical computation of ordinary differential equations,there is a class of new and efficient numerical solutions–boundary value methods and the induced block boundary value methods,which is proposed by Brugnano and Trigiante,two well-known mathematicians from Italy.Based on linear multistep methods,this type of methods not only has both good stability and excellent accuracy,but also can solve the implementation.Moreover,this type of methods can overcome Dahlquist' obstacle of order of linear multistep methods,and has both good stability and excellent computational accuracy.With the deeper expansion of this type of methods,they have been widely applied to solve the initial and boundary value problems of ordinary differential equations?linear differential algebraic equations?Hamilton problems?partial differential equations?Volterra integral differential equations and various differential equations with discrete and distributed delay.However,to the best of our acknowledge,block boundary value methods haven't been applied to nonlinear functional differential algebraic equations up to now.With view of it,the present paper will fill this gap and extend block boundary value methods to solve nonlinear functional differential algebraic equations with constant delay?nonlinear functional differential-algebraic equations with piecewise continuous arguments and nonlinear hybrid systems with distributed delay.In the end,combined with compact difference methods,a class of high-order compact block boundary value methods is extended to solve a class of semi–linear delay–reaction–diffusion equation with algebraic constraint.The framework of the present paper is organized as follows:Chapter 1 introduces application background and the research status of nonlinear functional differential algebraic equations.Then,we give a brief introduction about basic idea of underlying boundary value methods.Finally,we summarize research results of the present paper.Chapter 2 attacks the numerical solution of nonlinear functional differential algebraic equations with constant delay.Block boundary value methods are extended to solve functional differential and functional equations.Under some suitable conditions,it is shown that the extended block boundary value methods are uniquely solvable?globally stable and convergent of order p,where p is the consistency order of underlying boundary value methods.With several numerical examples,computational validity of the extended block boundary value methods is further confirmed.Chapter 3 deals with the numerical computation and analysis of a class of nonlinear functional differential algebraic equations with piecewise continuous arguments.Block boundary value methods are extended to solve this kind of equations.Under some appropriate conditions,it is proved that the extended block boundary value methods are uniquely solvable?globally stable and convergent of order p,where p is the consistency order of underlying boundary value methods.In the end,several numerical examples are provided to further confirm the computational effectiveness and theoretical results of the extended block boundary value methods.Chapter 4 investigates the numerical solution of a class of nonlinear hybrid systems with distributed delay,which include Volterra delay-integro-differential equations and Volterra delay-integral equations as special cases.Block boundary value methods,combined with reducible quadrature rules based on the underlying boundary value methods,are extended to solve this class of hybrid systems with distributed delay.Under some suitable conditions,it is proved that the extended block boundary value methods are uniquely solvable?globally stable and convergent of order p,where p is the consistency order of underlying boundary value methods.Finally,we provide one numerical experiment to further illustrate the corresponding theoretical results and computational effectiveness of the extended block boundary value methods.Chapter 5 is concerned with a class of high-order accurate scheme for semi-linear delay-reaction-diffusion equations with algebraic constraint.Then,a class of compact block boundary value methods,which combines fourth-order compact difference schemes for discretizing the spatial variable with block boundary value methods for discretizing the temporal variable,is extended to solve a class of semi-linear delay-reaction-diffusion equation with algebraic constraint.It is proved under some suitable conditions and rational hypothesis that compact block boundary value methods are globally stable and convergent of order4 in space and order p in time,respectively,where p is the consistency order of underlying boundary value methods.With several numerical experiments for Fisher equation with delay and algebraic constraint,the computational effectiveness and theoretical results of this class of compact block boundary value methods are further illustrated.In last chapter,we make a brief summary of the whole paper and list some open issues in the further.